Correct Answer - Option 4 : 38.5 m

__Concept:__

For calculating head loss or loss of pressures first need to find out Reynolds number. Reynolds number is used to find out type of flow.

\(Re = \frac{{\rho VD}}{\mu }\)

Reynolds number (Re) < 2000 flow is laminar

Reynolds number (Re) > 4000 flow is turbulent

If Reynolds number (Re) is 2000 to 4000 then flow is transit.

If Flow is laminar, we can use Hagen Poiseuille formula for calculating head loss which is given by

\(head\;loss\;\left( {{h_f}} \right) = \frac{{32\; \times \;\mu \;\times\; V\; \times\; L}}{{\rho \times g \times {D^2}}}\)

__Calculation:__

Given ρ = 1790 kg/m^{3}, μ = 2.1 Ns/m^{2}, V = 3 m/s, L = 12 m, D = 0.06 m;

\(Re = \frac{{1790\; \times \;3\; \times \;0.06}}{{2.1}}\)

\(Re = 153.43{\rm{\;}} < 2000\)

Flow is laminar so we can use Hagen Poiseuille formula for calculating head loss.

\({h_f} = \frac{{32\; \times \;\mu \; \times\; V\; \times \;L}}{{\rho \times g \times {D^2}}}\)

\({h_f} = \;\frac{{32 \;\times\; 2.1\; \times\; 3 \;\times\; 12}}{1790 \times9.81 \times {{{\left( {0.06} \right)}^2}}}\)

h_{f} = 38.26 m, nearest option is 38.5 m